exp2 (problem 3.3.7)

Average Error: 29.4 → 0.6

Time: 11.2s

Precision: binary64

Cost: 13568

Math

\[\left(e^{x} - 2\right) + e^{-x} \]

↓

\[\mathsf{fma}\left(x, x, {x}^{4} \cdot \left(0.002777777777777778 \cdot \left(x \cdot x\right) + 0.08333333333333333\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (fma
  x
  x
  (* (pow x 4.0) (+ (* 0.002777777777777778 (* x x)) 0.08333333333333333))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

↓

double code(double x) {
	return fma(x, x, (pow(x, 4.0) * ((0.002777777777777778 * (x * x)) + 0.08333333333333333)));
}

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

↓

function code(x)
	return fma(x, x, Float64((x ^ 4.0) * Float64(Float64(0.002777777777777778 * Float64(x * x)) + 0.08333333333333333)))
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * x + N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.4
Target	0.0
Herbie	0.6

\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \]

Derivation

1. Initial program 29.4

   \[\left(e^{x} - 2\right) + e^{-x} \]

2. Simplified 29.4

   \[\leadsto \color{blue}{-2 + \left(e^{x} + e^{-x}\right)} \]
   Proof
   (+.f64 -2 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= metadata-eval (neg.f64 2)) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (neg.f64 2) (exp.f64 x)) (exp.f64 (neg.f64 x)))): 1 points increase in error, 6 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 x) (neg.f64 2))) (exp.f64 (neg.f64 x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 x) 2)) (exp.f64 (neg.f64 x))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.6

   \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 0.6

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, 0.08333333333333333 \cdot {x}^{4}\right)\right)} \]
   Proof
   (fma.f64 x x (fma.f64 1/360 (pow.f64 x 6) (*.f64 1/12 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/360 (pow.f64 x 6)) (*.f64 1/12 (pow.f64 x 4))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/12 (pow.f64 x 4)) (*.f64 1/360 (pow.f64 x 6))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (+.f64 (*.f64 1/12 (pow.f64 x 4)) (*.f64 1/360 (pow.f64 x 6))))): 6 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (+.f64 (*.f64 1/12 (pow.f64 x 4)) (*.f64 1/360 (pow.f64 x 6)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 x 2) (*.f64 1/12 (pow.f64 x 4))) (*.f64 1/360 (pow.f64 x 6)))): 0 points increase in error, 1 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/360 (pow.f64 x 6)) (+.f64 (pow.f64 x 2) (*.f64 1/12 (pow.f64 x 4))))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in x around 0 0.6

   \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}}\right) \]

6. Simplified 0.6

   \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{{x}^{4} \cdot \mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right)}\right) \]
   Proof
   (*.f64 (pow.f64 x 4) (fma.f64 1/360 (*.f64 x x) 1/12)): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 x 4) (fma.f64 1/360 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 1/12)): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 x 4) (fma.f64 1/360 (pow.f64 x (Rewrite<= metadata-eval (/.f64 4 2))) 1/12)): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 x 4) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/360 (pow.f64 x (/.f64 4 2))) 1/12))): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 x 4) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 x (/.f64 4 2)) 1/360)) 1/12)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (pow.f64 x 4) (*.f64 (pow.f64 x (/.f64 4 2)) 1/360)) (*.f64 (pow.f64 x 4) 1/12))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 x 4) (pow.f64 x (/.f64 4 2))) 1/360)) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (*.f64 (pow.f64 x 4) (pow.f64 x (Rewrite=> metadata-eval 2))) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (*.f64 (pow.f64 x 4) (Rewrite=> unpow2_binary64 (*.f64 x x))) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (pow.f64 x 4) x) x)) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (*.f64 (Rewrite=> pow-plus_binary64 (pow.f64 x (+.f64 4 1))) x) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite=> pow-plus_binary64 (pow.f64 x (+.f64 (+.f64 4 1) 1))) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (pow.f64 x (+.f64 (Rewrite=> metadata-eval 5) 1)) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (pow.f64 x (Rewrite=> metadata-eval 6)) 1/360) (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (pow.f64 x 6) 1/360) (Rewrite<= *-commutative_binary64 (*.f64 1/12 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/360 (pow.f64 x 6))) (*.f64 1/12 (pow.f64 x 4))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 0.6

   \[\leadsto \mathsf{fma}\left(x, x, {x}^{4} \cdot \color{blue}{\left(0.002777777777777778 \cdot \left(x \cdot x\right) + 0.08333333333333333\right)}\right) \]

8. Final simplification 0.6

   \[\leadsto \mathsf{fma}\left(x, x, {x}^{4} \cdot \left(0.002777777777777778 \cdot \left(x \cdot x\right) + 0.08333333333333333\right)\right) \]

Alternatives

Alternative 1
Error	0.7
Cost	13184

\[\mathsf{fma}\left(x, x, {x}^{4} \cdot 0.08333333333333333\right) \]

Alternative 2
Error	1.1
Cost	192

\[x \cdot x \]

Alternative 3
Error	60.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))